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SAT Conditional Probability Two-Way Tables

Most SAT probability questions come with a two-way table of counts and a phrase like given that or among those who. That phrase shrinks the group you are choosing from, which changes the denominator of the fraction. Picking the denominator correctly is essentially the entire question.

The rule

The phrase after given that or among names your denominator, which is a row or column total rather than the grand total.

A probability is the count of outcomes that satisfy the condition, divided by the count in the group you are selecting from. In an unconditional question that group is everyone, so the denominator is the grand total of the table. Nothing else about the table matters for that case.

A conditional question restricts the group first. Suppose a table splits 200 students by grade and by whether they take a language. If 80 students are juniors and 50 of those juniors take a language, then the probability that a randomly chosen junior takes a language is 50 over 80, which is five eighths. The 200 never enters the calculation, because you are only choosing among juniors.

Order matters in these phrases and the test exploits that. The probability that a randomly selected student is a junior who takes a language is 50 over 200. The probability that a randomly selected junior takes a language is 50 over 80. The probability that a randomly selected language student is a junior would use the language column total as the denominator. Same table, same cell on top, three different answers.

So read the sentence in two passes. First find the restricting phrase, which is usually the words after given that, among, or of those. Circle that row or column total, because it is your denominator. Then find the single cell that satisfies the remaining condition, and that is your numerator.

Some questions run this backward, giving you a probability and asking for a missing count in the table. Set up the same fraction with a letter in the unknown spot and solve. Before you finish, check that your filled-in table still has rows and columns that add to their stated totals, which catches a misplaced value immediately.

The trap

Using the grand total as the denominator on a question that restricted the group to one row or column.

Try it

Try one
Real question

A fair 12-sided die with faces numbered 1 through 12 is rolled once. What is the probability of rolling a number that is a multiple of 3?

Of 500 people surveyed, 200 said they own at least one pet. Of the pet owners, 25% own more than one pet. If one of the 500 people is selected at random, what is the probability that the person owns exactly one pet?

  • A) 0.1
  • B) 0.25
  • C) 0.3 ✓
  • D) 0.4

Why: Of the 200 pet owners, 0.25(200) = 50 own more than one, so 150 own exactly one. Out of all 500 people that is 150/500 = 0.3. Answering 0.4 counts every pet owner, including those with more than one pet. Answering 0.1 means you gave the probability that the person owns more than one pet. Answering 0.25 means you gave the share of pet owners with more than one pet rather than a probability over all 500 people. Answering 0.4 means you gave the probability that the person owns at least one pet.

Practice more of these free

Questions students ask

How do I spot a conditional probability question?

Look for wording that narrows the pool before the selection happens, such as given that the person is a senior, among those who responded yes, or of the students who chose option B. Any of these means the denominator is that subgroup total, not the table total.

What is the difference between probability of A and B versus probability of A given B?

Probability of A and B counts people in both categories out of everyone, so the denominator is the grand total. Probability of A given B counts the same people out of only the B group, so the denominator is the B total. The numerator is the same cell in both cases.

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